Is it true that if $$ \sum_{k=1}^{\infty}{a_k^nt_k}=0 $$ for all $ n\in\mathbb{N}\cup\{0\} $ and some fixed sequence $ \{a_k\}_{1\le k\le \infty},\,a_k\ne a_l \ \forall k\ne l $, then $ t_k=0 \ \forall k $? Do conditions like $ 0\le a_k \le 1$ and $ -1\le t_k\le 1 $ change the answer?
Edit. I understood that there is an interesting reformulation of this question: to prove (or refute) that this infinite system of linear equations $$ \begin{pmatrix} 1 & 1 & 1 & \dots & 1 & \dots \\ a_1 & a_2 & a_3 & \dots & a_n & \dots \\ a_1^2 & a_2^2 & a_3^2 & \dots & a_n^2 & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ a_1^k & a_2^k & a_3^k & \dots & a_n^k & \dots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \end{pmatrix}\begin{pmatrix} t_1 \\ t_2 \\ t_3 \\ \vdots \\ t_n \\ \vdots \end{pmatrix}=\begin{pmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ \vdots \end{pmatrix} $$ has no nontrivial solutions for $ t_i $. Although probably linear algebra isn't applicable here, since left matrix doesn't determine linear transformation due to divergence of series.